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Comments on certain divergence‐free tensor densities in a 3‐space

### Abstract

It is well known that a necessary and sufficient condition for the conformal flatness of a three‐dimensional pseudo‐Riemannian manifold can be expressed in terms of the vanishing of a third‐order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton’s tensor density is not the Euler–Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics.

© 1979 American Institute of Physics

Published online 29 July 2008

/content/aip/journal/jmp/20/9/10.1063/1.524289

1.

1.D. Lovelock, “Divergence‐Free Third Order Concomitants of the Metric Tensor in Three Dimensions,” in Topics in Differential Geometry, edited by H. Rund and W. F. Forbes (Academic, New York, 1976), p. 87.

2.

2.For notation see S. J. Aldersley, Phys. Rev. D 15, 370 (1977).

2.The definition of a tensorial concomitant used here is that on p. 35 of Ref. 9 in G. W. Horndeski, Utilitas Math. 9, 3 (1976). For example, a scalar density concomitant of a pseudo‐Riemannian metric is defined by a real‐valued function φ of the form The concomitant φ is said to be of class provided the function φ is of class at every metric in its domain. (For example, and are of class but is only continuous since it is not differentiable in any neighborhood of a metric for which at some point).

3.

3.E. Cotton, “Sur les varietés à trois dimensions,” Ann. Fac. d. Sc. Toulouse (11) 1, 385 (1899).

4.

4.J. W. York has used this tensor density with reference to the initial‐value problem in general relativity. See Phys. Rev. Lett. 26, 1656 (1971),

4.and C. W. Misner, K. S. Thome, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 541 and 550.

6.

6.G. W. Horndeski, Tensor 28, 303 (1974).

7.

7.I. M. Anderson Aeq. Math, (to be published). Also see Ref. 11.

8.

8.K. Kuchař [J. Math. Phys. 15, 708 (1974)] has considered the tensor density and shown that it is not an Euler‐Lagrange expression. [Note that is not symmetric unless one is constrained to metrics for which Hence the fact that for any L is obvious.] His method of proof does not apply to the Cotton tensor density in view of Eqs. (4) and the existence of a Lagrangian of the form (5). (That is, the functional curl of the Cotton tensor density vanishes identically).

9.

9.T. Y. Thomas, The Differential Invariants of Generalized Spaces (Cambridge U.P., Cambridge, 1934), p. 109.

10.

10.H. Weyl, The Classical Groups (Princeton U.P., Princeton, 1939), p. 52 noting p. 65.

11.

11.These points are discussed further in S. J. Aldersley, “High Euler Operators and some of their Applications,” preprint available from the author.

12.

12.J. W. York, J. Math. Phys. 14, 456 (1973).

13.

13.Generalizations of the Cotton tensor density for higher dimensional spaces will be discussed in S. J. Aldersley and G. W. Horndeski (in preparation).

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